38 Trans. Acad. Sci. of St. Louis. 



Draw a zigzag diagonal as shown dividing the rectangle 

 of numbers into two triangles of numbers. Divide the left- 

 hand triangle of numbers into rows ; divide the right-hand 

 triangle of numbers into columns. It will be evident by 

 inspection that the sum of the numbers in each column 

 of the right-hand triangle is equal to the sum of the num- 

 bers in the corresponding row of the left-hand triangle; 

 and that, therefore, the sum of all the numbers in the left- 

 hand triangle is equal to the sum of all the numbers in 

 the right-hand triangle, or equal to one-half the sum of all 

 the numbers in the whole rectangle. 



This will be true for a rectangle of any number of rows. 



The sum of the numbers in each row of the left-hand 

 triangle may be regarded as a term in a series of num- 

 bers, which, beginning at the bottom, is the series of nat- 

 ural numbers, 



1, 2, 3, 4, n, 



where n represents the highest term of the series under 

 consideration; that is ^ = 1, 2, 3, 4 . . ., depending upon 

 how far the triangle is extended. In the diagram drawn 

 n = 6. n also represents the number of the last column, 

 represented in the left-hand triangle, counting from the 



left. 



Now the sum of all the numbers in the whole rectangle 

 of numbers is equal to the sum of the numbers in each 

 row multiplied by the number of rows. 



The sum of the numbers in- each row of the rectangle 

 is n-\-l ; the number of rows is n. 



Therefore the sum of all the numbers in the whole 

 rectangle will be 



And the sum of all the numbers in the left-hand tri- 

 angle will be 



n(l^+l) 



S = 



1-2 



That is to say, the sum of the series 1, 2, 3, 4 . . . n is 

 S^'^^, (2) 



